Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#49667 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(11,12,14,16)(13,15,17,18)(19,20)(21,28)(22,30)(23,26)(24,25)(27,32)(29,33)(31,34);(4,7)(5,9)(19,21,25,23)(20,26,24,28)(22,31,32,29)(27,34,30,33);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,22,25,32)(20,27,24,30)(21,29,23,31)(26,33,28,34);(3,4)(5,8)(6,7)(9,10)(11,13,14,17)(12,15,16,18)(19,23,25,21)(20,26,24,28)(22,29,32,31)(27,34,30,33);(3,5)(4,8)(6,9)(7,10)(19,24,25,20)(21,26,23,28)(22,30,32,27)(29,33,31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612619);